Ordinary and degenerate Euler numbers and polynomials

Abstract

Abstract In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on Zp $\mathbb{Z}_{p}$. Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

Keywords

• Euler polynomials and numbers
• Degenerate Euler polynomials and numbers
• Alternating integer power sum polynomials
• Degenerate alternating integer power sum polynomials